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Some are rather volatile, and others remain rather stable in their state. The focus of research question 4 is to reveal if there are several typical patterns of observed behaviors that allow for identifying groups of couples that differ in their stress treatment. Detecting unobserved groups with different response patterns in categorical data is commonly done relying on Latent Class Analysis e. Hence, a first approach to answer research question 4 is to combine Latent Class Analysis and Markov modeling resulting in mixture Markov models Van de Pol and Langeheine, A second approach is sequence clustering Abbott, Mixture models or so-called latent class models assume that the population consists of several latent unknown subgroups.
Detecting these subgroups accounts for so-called unobserved heterogeneity in the population. Dealing with sequence data, the notion of unobserved heterogeneity implies that there are different subgroups differing in their specific transition matrices. The assignment of the couples to the latent classes is probabilistic, that is, for every couple, there are as many probabilities to belong to a particular latent class as there are classes.
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The couple is presumed to belong to the class with the highest assignment probability. As a special case, a mixture Markov model with only one latent group is equivalent to the basic Markov model, hence model comparisons between models without unobserved heterogeneity and models with multiple classes are possible. However, to illustrate mixture Markov models, results of the model with two latent classes are depicted in Table 9.
The latent classes contain 59 vs. The advantage of this model is that it accounts for unobserved heterogeneity caused by a categorical variable. A disadvantage is that, to our knowledge, no recommendations for required sample sizes exist. However, as a simulation study by Dziak et al. A second modeling approach is based upon the idea of subgrouping sequences.
Defining A Group
This modeling tradition is known as sequence analysis. However, we will refer to this approach as sequence clustering , because in this approach classical cluster analysis is adapted for sequence data.
Within this approach, optimal matching procedures OM; Abbott and Tsay, can be seen as a viable alternative to mixture Markov models. Essentially, OM categorizes behavioral sequences of individuals or couples according to their similarity in a stepwise procedure: The first step is defining dis- similarity via an appropriate distance measure in OM called cost , the second step is identifying clusters of similar sequences by applying a clustering algorithm.
These clusters can be interpreted, and covariates may be included in a statistical model predicting cluster membership.
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In the first step, the metric of similarity and difference between sequences is defined by Levenshtein -distances : Two sequences are similar if essentially the same pattern of behavior is shown Levenshtein They differ to the extent to which some elements of one sequence have to be changed to perfectly match the other sequence cost. In this case, the two sequences can be made identical by removing the first element in the first sequence and shifting the remainder of the sequence to the left deletion , or by copying the first element of couple 1 and paste it at the beginning of the second sequence insertion.
Therefore, the minimal cost of transforming both sequences into each other is one operation. Consider a second case with two totally identical sequences which only differ at the entry in the fourth interval. Again only one operation is needed. However, substitution is weighted differently than insertion or deletion, and the cost of this transformation would be one times a weight weighting. A higher weighting stands for more dissimilarity. That is, a basic Markov model is fitted, and the transition probabilities of two states, which should be substituted, are subtracted from two.
Thus, if the transition between the two states is very likely, the weighting becomes smaller. The value becomes zero when each of the two states is always followed by the other. And if they never occur in consecutive order, the weight becomes two. For every two observation units couples the minimal cost is computed for transforming their sequences into each other. The results are stored in a distances matrix with as many rows and columns as number of observations. Cells represent the minimal cost between the associated observation units.
This dissimilarity matrix corresponds to other distance measures, like the Euclidian distance for example, in an ordinary cluster analysis, except that it assumes sequence data rather than metric data. In a second step, clusters of similar sequences can be identified via a clustering algorithm. In this example, the Ward error sum of squares hierarchical clustering method Ward, will be used because that is the default algorithm in the R-Package TraMineR Gabadinho et al. The algorithm is commonly used Willett, , yields a unique and exact hierarchy of cluster solutions, and is comparable to most methods for identifying the number of clusters.
The algorithm treats each sequence as a single group, in the beginning. Then pairs of sequences are merged stepwise minimizing the within-group variances. The latter is determined by the squared sum of distances between each single observation and its clusters centroid. The silhouette test Kaufman and Rousseeuw, can be used for determining which cluster solution provides the best representation of the data. The test is well-established and yields the benefit that it provides the silhouette coefficient reflecting the consistency of clusters. According to Struyf et al.
In the application, the silhouette test resulted in a two-cluster solution with a coefficient of 0. Hence, additional methods should be used for validating the findings. These may include inspection of the dendrogram, scree plot, or principle component plot. All three methods lead to a two-cluster solution see the accompanying R-script.
The last step is to interpret or to describe the clusters. Figure 6 shows the state-distribution plots for both clusters. The obvious difference between the two clusters is that, in cluster 1, couples quickly enter states of no stress communication and no dyadic coping, whereas, in cluster 2, most of the couples remain in the state of stress communication and dyadic coping over the whole interaction sequence. Figure 6.
State distribution plots both clusters identified by the OM-procedure. Covariates can be used for further interpretations. Additional possible follow-up investigations include the application of strategies from previous sections. Table 10 provides the results of applying the aggregated logit models separately for both clusters. This finding indicates that the DC response for this group is less likely or not as prompt.
Furthermore, the interaction effect was not statistically significant in the overall sample. These findings indicate that at least two separate styles of dyadic coping might exist. Table Mean log-linear parameter comparison for cluster 1 and 2 with DC as depended variable. Identifying the exact nature of these separate styles might be subject of further research. An alternative explanation might be that stressed partners of slow coping couples like to be comforted and keep their SC up so that their partners keep up their DC. The main purpose of this article is promoting the presented analyses.
Thus, a detailed substantive discussion of the findings with respect to couple research will not be provided. Instead, the overall findings will be sketched in a more general way to highlight the main interpretations of the different statistical models. The Pearson correlation revealed a strong linear relationship between the number of observed SC and DC. Aggregated logit model and the multi-level model revealed the bidirectional effects between these two variables.
The multi-level model also revealed that couples with high actor effects also tend to show higher partner effects. However, the nature of this relationship seems to be different across couples.
summit.vvinners.com/cylo-jenbacher-manual.php The entropy plot shows that couples are similar at the beginning of the observation period but start to differ at interval 20 2 min; 40 s. Sequence clustering revealed that the sample can be clustered into two groups, while the mixture Markov model did not reveal two classes. At first glance, this seems somewhat inconsistent. However, the OM-procedure assigned The first cluster can be characterized by a faster increase of the state without SC and DC.
Additionally, this cluster fast coper shows a shorter duration of SC while the other slow coper shows much longer durations and higher rates of SC and DC states. These findings may be used for further research questions: For example, it could be interesting to investigate if belonging to the fast vs.
A commented R-script for reproducing all results of this paper can be downloaded at GitHub 1. The aggregated logit model, the multi-level model and the presented Markov models assume stationarity. According to Helske and Helske such models can still be useful for describing data, even if stationarity cannot be assumed. However, Markov models can be extended to semi-Markov models Yu, , which do not rely on this assumption.
A disadvantage of our multi-level adaptation is that the model does not allow for estimating random effects for both members of a dyad separately. Kenny et al. Sequence clustering is very flexible and can be configured in many different ways. Therefore, it is important to justify specifications of the statistical model or notify if the specifications were chosen arbitrarily. Thus, it uses the same information as the aggregated logit models, and therefore, is best suited to detect subgroups with respect to prompt reactions.
However, alternative methods for deriving the substitution-cost-matrix can be found in Gauthier et al. Alternatives also exist regarding the clustering algorithm. An extensive overview of alternatives can be found in Kaufman and Rousseeuw However, one of the most appealing features of the presented Ward algorithm is that it is comparable to many other methods for determining the correct number of clusters. In some cases different tests may indicate different numbers of clusters.
In these cases, additional analyses with additional variables including cluster membership as a covariate may reveal if the clusters differ from another. Meaningful differences between association patterns for these additional variables between clusters may indicate that these clusters exist. However, OM-procedures as cluster analysis are explorative in nature, and findings should be cross-validated before being generalized.
It is worth to mention that mixture Markov models can be combined with any other Markov modeling approach as well. For example, it is also possible to apply a mixture hidden Markov model MHMM resulting in different common fate models for latent classes. Moreover, the R-package seqHMM Helske and Helske, provides a multi-channel approach to estimate separate emissions for each dependent variable. Dropouts may occur meaning that observation units leave the sample before the observation period has ended. Furthermore, they may produce sequences with different lengths.